Non-Newtonian central force and orbit precession

• The unit mass m is the particle mass (or the reduced mass for the 2-body system).
• The unit length is the radius of the circular orbit in the attractive Newtonian case: r₀ = L²/m|k|.
• The unit time is L³/mk².
• In these units we have m = L = |k| = 1, orbit's equation is 1/r = ε cos ω(φ-φ₀) + η, with η = |k|/k, ε = [1+2(1+σ)E]^1/2 and ω = (1+σ)^1/2.

• On the left appears in red the effective potential energy -η/r+(1+σ)/2r²: its maximum is located at the point (1,-(1+σ)/2) in the attractive case. You may use the mouse (or the controls below) to select the mechanical energy E and the initial polar distance r.
• Particle's plane motion (or the relative motion in the 2-body problem) is displayed on the right. You may use the mouse to select the initial position: the program will automatically set the orbit orientation, r and, if necessary, E.
• Put the mouse pointer over an element to get the corresponding tooltip.

Activities

• Set σ=0 (sigma in the simulation) and make sure you recover the Newtonian orbits.
• Set a small value for σ and choose a negative energy. Check that the orbit is still bounded but, in general, open (i.e., non periodic) for the pericentron (or the apocenter) does not happen always in the same direction. For small σ (say 0.02) you should get something like the first orbit in the following image:

  

• Use the orbit equation to proof that, for E < 0, the orbit is closed if and only if ω is rational, i.e., if one can write σ = p²/q²-1 for rational p, q, = 1,2,... (This result is obviously compatible with Bertrand's theorem that states that if all bounded orbits are closed the force is Newtonian or harmonic.)
• Check the theoretical result in the simulation. In particular, you should be able to reproduce the remaining three orbits in the figure.
• If the orbit goes very near the force center, you may want to uncheck Use L, so that the animation pace slows down when the particle is near the center: this is just the opposite to what should happen (as a consequence of the angular momentum conservation) in real time, but the resulting orbit should look better.